Symmetries of the pseudo-diffusion equation and related topics

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We show in details how to determine and identify the algebra g = {Ai} of the infinitesimal symmetry operators of the following pseudo-diffusion equation (PSDE) LQ ≡ [∂∂t−14(∂2∂x2−1t2∂2∂p2)]Q(x, p, t) = 0. This equation describes the behavior of the Q functions in the (x, p) phase space as a function of a squeeze parameter y, where t = e2y. We illustrate how Gi(λ) ≡ exp[λAi] can be used to obtain interesting solutions. We show that one of the symmetry generators, A4, acts in the (x, p) plane like the Lorentz boost in (x, t) plane. We construct the Anti-de-Sitter algebra so(3, 2) from quadratic products of 4 of the Ai, which makes it the invariance algebra of the PSDE. We also discuss the unusual contraction of so(3, 1) to so(1, 1)∌ h2. We show that the spherical Bessel functions I0(z) and K0(z) yield solutions of the PSDE, where z is scaling and “twist” invariant.

Original languageEnglish
Pages (from-to)334-339
Number of pages6
JournalPhysics of Atomic Nuclei
Issue number2
StatePublished - 1 Mar 2017

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Nuclear and High Energy Physics


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